Maths refresher course for Economics

1. Maths refresher course for Economics Part 1: Why economics contains so much maths

2. Part 1 Why economics contains so much maths The Scientific approach What is a model ?

3. Why economics contains so much maths • Economics tries to understand the behaviour of agents • Resources are limited, therefore agents have to make choices. • These depend on the incentives faced by the agent • Because agents are different, they can benefit from exchange • Producers and consumers therefore meet on markets that ensure an efficient use of these resources

4. Why economics contains so much maths • The aim of economics if to answer the following questions: • What to consume ? How much ? • What to produce ? (Which good ?) • How to produce it ? (Which technology ?) • Why are some countries richer than others? • Why do some countries have high unemployment ? High inflation?

5. Why economics contains so much maths • In order to answer these questions and understand how agents make their decisions, economics models: • The decision making process of the agent • The flows of goods and services in the economy • We use models because it is impossible to understand directly the depth and complexity of human behaviour • This is what we’ll examine in the next 2 sections, and look at how models are used in science

6. Why economics contains so much maths • Example of Sir Arthur Lewis (Nobel prize 1979) on Dani Rodriks’ Blog • http://rodrik.typepad.com/dani_rodriks_weblog/2007/09/why-we-use-math.html • There is an apparent paradox: • « Maths are complicated » • In fact, it’s a simplification !! • In other words, we use maths because we’re not intelligent enough to do without !!

7. Why economics contains so much maths • Mathematics allows us: • To use a symbolic notation for all the variables in a given problem. • To develop notations and rules that define logical relations (logical operations are “codified”) • As a result one can carry out a complex series of logical operations without making mistakes, forgetting variables, etc.

8. Why economics contains so much maths • Which types of mathematics do economists use ? • Algebra and Calculus, mainly analysing the properties of functions • Part 2 • Statistics • Which you will see with Evens Salies on Friday

9. Part 1 Why economics contains so much maths The Scientific approach What is a model ?

10. The Scientific approach • What makes something “Scientific” ? • A lab coat ? • Laboratory equipment? • The capacity to run experiments ?

11. The Scientific approach • The central objective of science : explain the phenomena that we observe • In other words we try and understand the causal links of a problem • Understanding a problem is finding its cause ! • Practical aspect: How do we do this in a complex world? • Several different explanations are possible • They can also interact!

12. The Scientific approach • You need a systematic method to evaluate all the possible explanations, and eliminate those that are not valid. • In particular you need to be able to impose a « ceteris paribus » condition (all other things being equal) • Example of the thermometer and temperature • Therefore you need to be able to create a simplifiedrepresentation of reality to be able to isolate these effects! • We will see that this is where the maths come into play

13. The Scientific approach • The scientific method: Theory Reality You identify variables that can explain it You observe a phenomenon No: you start over again! You write a model (simplify the problem) Do the predictions fit the data? You get a set of predictions Yes: you have a valid theory

14. Part 1 Why economics contains so much maths The Scientific approach What is a model ?

15. What is a model? • What is a model ? • “A simplified representation of reality” • In other words, a representation which removes the unnecessary complexity of reality to focus on the key mechanisms of interest • “A model’s power stems from the elimination of irrelevant detail, which allows the economist to focus on the essential features of economic reality.” (Varian p2)

16. What is a model? • It is important to understand that models are central to how humans perceive reality • Human understanding of the world (not just in economics !) comes from understanding simplified versions of a complex world. • The role of the scientific process is to separate good and valid simplifications from invalid ones. • “One must simplify to the maximum, but no more” Albert Einstein

17. What is a model? • Illustration of a general, simple “model” • You are in Nice • You don’t know your way around, and you get lost. • You ask a passerby where you are • This person gives you two possible answers as to your location • Which is the more useful (i.e. instructive model) ?

18. What is a model? You are here

19. What is a model? You are here

20. What is a model? • Modelling in economics • We assume a simplified agent and environment • even if you know that this is unrealistic !! • We try and understand how things work in this ideal situation. • Then we try and relax the simplifying assumptions one by one and see how the mechanisms change

21. What is a model? • The simplified agent used is typically called the “Homo œconomicus” • Has complete knowledge of his objectives (preferences or production quantities) • Has complete knowledge of the conditions on all the markets (perfect information) • Has a very large “computational capacity” to work out all the possible alternatives and their payoffs. • These simplifications can be relaxed

22. Maths refresher course for Economics Part 2: Basic Calculus

23. Part 2 What is a function ? Calculus and optimisation The derivative of a function Constrained maximisation

24. What is a function ? “ Many undergraduate majors in economics are students who should know calculus but don’t – at least not very well” (Varian, preface) So before starting on the models and the theory, it is important to understand the components of models : functions

25. What is a function ? • A function is a relation between: • A given variable that we are trying to explain • A set of explanatory variables • A variable is a quantity: • That varies with time, • That can be measured on a given scale • Examples: Temperature, pressure, income, wealth, age, height

26. What is a function ? • The relation between two variables X and Y can be: • Positive (or increasing): • Variations happen in the same direction • Negative (or decreasing): • Variations happen in opposite directions

27. What is a function ? • Practical Example : Let’s use a road safety example • You are asked by the Ministry of the Interior to identify a cost-effective way of reducing the number of deaths on the road due to car accidents. • Which are the important variables? • What measures are associated ? • Is the direction of the relation?

28. What is a function ? • The same function can have different “faces” • The same relation between variables can be expressed in different ways • 1: “Literary” representation • This is the one from the previous slide, and involves just mentioning the variables that enter the function • “The number of accidents is a positive function of averagerainfall, the speed of driving and the quantity of alcohol consumed.”

29. What is a function ? • 2: Symbolic representation • A bit more “rigorous”, this uses symbols to represent the relation between variables Mathematical symbol meaning “function of” • Where a is the number of accidents, ris rainfall, s is the speed and q is alcohol consumption. • But... When read out, this just corresponds to the literary version !!

30. What is a function ? • 3: Algebraic representation • This is the “scary” one, because it involves “maths” (algebra, actually) • The problem is that to express a function this way, you need to know exactly: • The “functional form” (Linear, quadratic, exponential) • The values of the parameters • Finding these is often part of the work of an economist

31. What is a function ? a (accidents /year) Car accidents as a function of rainfall r (cm/m2) • 4: Graphical representation • Often the most convenient way of representing a function...

32. What is a function ? a (accidents /year) Car accidents as a function of rainfall q1>q r (cm/m2) • ... But a diagram can only represent a link between two variables (aand r here) • If alcohol consumption q increases, then a whole new curve is needed to describe the relation

33. Part 2 What is a function ? Calculus and optimisation The derivative of a function Constrained maximisation

34. Calculus and optimisation • The economic approach often models the decision of an agent as trying to choose the “best” possible outcome • The highest “satisfaction”, for consumers • The highest profit, for producers • Imagine a function f that gives satisfaction (or profits) as a function of all the quantities of goods consumed (or produced).

35. Calculus and optimisation satisfaction Maximum Graphically, that’s easy! But generally, how do you find this maximum ? q In terms of modelling, finding the “best choice” is effectively like trying to find the values of the quantities of goods for which function f has a maximum

36. Calculus and optimisation • For both examples, the optimum is the point where the function is neither increasing nor decreasing: • Satisfaction no longer increases but is not yet falling. • Road deaths are no longer falling but aren’t yet increasing. • This is basically how you find maxima and minima in calculus. • The methods may seem ‘technical’, but the general idea is simple

37. Part 2 What is a function ? Calculus and optimisation The derivative of a function Constrained maximisation

38. The derivative of a function • Imagine that we want to find the maximum of a particular function of x • Example • How do we find out at which point it has a maximum without having to use a graph? • We need to introduce the concept of a derivative

39. The derivative of a function • We will use the following approach: • We will introduce the concept of a tangent • This will allow us to introduce the concept of a derivative using the graphical approach, which is more intuitive. • You are on a given point on a function, and you want to calculate the slope at that point

40. y Δx = h x The derivative of a function y2 = f(x + h) y = f(x) Δy = y2 – y1 y1 = f(x) x x + h

41. y x The derivative of a function Problem: Different points give different slopes Which one gives the best measurement of the slope of the curve?

42. y x The derivative of a function Mathematically, the measurement of the slope gets better as the points get closer. The best case occurs for aninfinitesimalvariationin x. The resulting line is called thetangent.

43. y x The derivative of a function The tangentof a curve is the straight line that hasa single contact pointwith the curve, and the two form a zero angle at that point.

44. y x The derivative of a function The slope of the tangent is equal to the change in y following aninfinitesimalvariationin x. Δy Δx

45. y x The derivative of a function As we saw, a maximum is reached when the slope of the tangent is equal to zero.

46. The derivative of a function y Slope = 0 B C A Slope > 0 Slope < 0 1 1 -s s x

47. The derivative of a function • With continuous functions, each curve is made up of an infinite number of points • This is because points on the curve are separated by infinitely small steps (infinitesimals) • There is an infinite number of corresponding tangents • This is not the case for discrete curves • There is only a finite number of points (no infinitesimals) • What we need is a recipe for calculating the slope of a function for any given point on it • Luckily, even though there are an infinite number of points, this allows us to derive such “recipes”.

48. y Δx = h x The derivative of a function y2 = f(x + h) y = f(x) Δy = y2 – y1 y1 = f(x) x x + h

49. The derivative of a function • General rule: • Let fbe a continuous function defined at point. The derivative of the function f’(x) is the following limit of function f at point x : • In other words, it is literally the calculation of the slope as the size of the step become infinitely small. • This is done using limits, but we will use specific rules that don’t require calculating this limit every time

50. The derivative of a function